Let $x_i=p_1^{t_1}p_2^{t_2}...p_m^{t_m}$ Let $a_n=(x_n \pmod{p_1},x_{n+1} \pmod {p_1})$ is sequence of pairs. $a_n$ can take not more than $p_1^2$ different values and so it is periodic with period $T_1$ So $a_i=a_{i+kT_1} \to x_{i+kT_1} \equiv x_i \equiv 0 \pmod {p_1}$ for positive integer $k$ Then we can find such $T_1,T_2,..,T_m$ that $p_m|x_{i+kT_m}$ for $n \leq m$ and positive integer $k$ Let $T=T_1...T_m$ then $p_1p_2...p_m|x_{i+kT}$ for positive integer $k$ we can choose big enough $k$ such that $j=i+kT \geq i*max (v_{p_n}(x_i))$( $n \leq m$) and then we will have $v_{p_n}(x_j^j) \geq j \geq i*max (v_{p_n}(x_i))$ and so $x_i^i|x_j^j$

What an original problem! See ISL 1994 N6 https://artofproblemsolving.com/community/c6h31735p196779

Chosing problem for National MO from different contests maybe understandable, but choosing from old SL is very absurd...

lmao goofy ah problem